Quantized Polarization Redefines Polar Interfaces

In crystalline solids, the electronic polarization follows the \emph{generalized Neumann's principle}, under which all crystallographic point groups can, in principle, support ferroelectric polarization. However, in high-symmetry structures, polarization is constrained by symmetry operations and becomes quantized into discrete values. We demonstrate that this quantized polarization (QP) is not a mathematical artifact but a \emph{symmetry-protected invariant} that encodes intrinsic information about a material's symmetry and electronic structure. Because of its discrete and non-continuous nature, when two materials with different QPs form an interface, their bulk polarization states cannot be connected adiabatically, compelling the system to develop pronounced interfacial responses: such as metallic states, bound charges, or strong lattice distortions. This theoretical framework provides a unified reinterpretation of classical systems such as the LaAlO$_3$/SrTiO$_3$ interface, revealing it as a prototypical case of QP mismatch. By establishing QP as a fundamental bulk invariant, our work uncovers a universal mechanism governing interfacial electronic phenomena and opens new pathways for the design of functional quantum materials through engineered polarization mismatch.